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\title{ODE}
\author{Ricardo Cruz}

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\section{Introduction}

\citet{Perelson1993} worked out a first model of HIV dynamics using a system of \textbf{\glspl{ODE}}. The model tracked: healthy helper T cells ($T$), latently infected helper T cells ($T^*$), productively infected helper T cells ($T^{**}$), and free virions ($V$). Usually, productively infected and latently infected are merged to a single infected helper T cell compartment ($T^*$), as they simplified in a later article \citep{Perelson1996}. Even the simple \gls{ODE} described below can provide for a statistically significant fit of patients disease history, and thus find some interesting parameters and properties about their disease. A system of \glspl{ODE} is used to model the disease, whereby compartments are used to represent the various agents, in their various states. These models may even carry spatial information \citep{Graw2012}. Some examples are presented below. \emph{(See section \ref{ODE}.)}

\glspl{ODE} can represent discrete properties of each individual being model, through the usage of compartments, but fails short when it comes to continuous variables such as physical coordinates for spatial modeling, which must be discretized. \textbf{\glspl{PDE}} allow the incorporation of continuous variables, and, for instance, \citet{Su2009} makes use of \glspl{PDE} to attempt at a model of spacial immune response. Of course, numerical integrators will discretize the system of \glspl{PDE}.




\section{Fundamental Model}

One of the first HIV models is an ODE by \citet{Perelson1993}. \footnote{I have it implemented in NetLogo at: \url{https://www.dropbox.com/s/2ef4vmamjj5wj8w/eqdiff.nlogo?dl=0}} A slightly stripped down and simplified version of their initial model can be rendered as:

\begin{align*}
\frac{\text{d}T}{\text{d}t} &= \lambda - d T - \beta VT \\
\frac{\text{d}T^*}{\text{d}t} &= \beta V T - \delta T^* \\
\frac{\text{d}V}{\text{d}t} &= N \delta T^* - c V
\end{align*}

\noindent
where
\vspace{-\topsep}
\begin{itemize}
\item[$\lambda$] is the constant rate of helper T cell production by the thymus,
\item[$d$] is the natural rate of death,
\item[$\beta$] is the rate at which helper T cells become infected as they interact with free virions,
\item[$\delta$] is the rate at which incubators/infected helper T cells burst,
\item[$N$] is how many free virions are released on burst, and
\item[$c$] is the rate at which free virions die or are killed.
\end{itemize}
\vspace{-\topsep}

\noindent
\citep{Perelson1996,Nowak1996}

This model does not reproduce the three infection stages of HIV; it refers to the ongoing disease. For the actual reproduction of the stages, delay differential equations are used to model the supposed latency period of HIV: \citep{Culshaw2000}

\begin{align*}
\frac{\text{d}T}{\text{d}t} &= \lambda + d T - \beta V(t)T(t) \\
\frac{\text{d}T^*}{\text{d}t} &= \beta V(t-\tau)T(t-\tau) - \delta T(t)^* \\
\frac{\text{d}V}{\text{d}t} &= N \delta T(t)^* - c V(t)
\end{align*}

(To do: is this enough to mimic the three stages of infections?) Furthermore, work is done generalizing $\tau$ so that, instead of a constant, it follows distribution $f(\tau)$. \citep{Nelson2002}

Nowak's model introduced killer T cells and mutations into the mix, and managed to replicate the three HIV infection phases in 1996. The model is actually similar in concept to the cellular automata described below (\ref{ABM}). Notice $i$ and $j$ are strains. He had to use 10 strains in order to model the three stages.

\begin{align*}
\frac{dx_i}{dt} &= \nu c_i v_{i*} + x_i(c_iv_{i*}-b), \quad i=1,\dots,n_1& \\
\frac{dy_j}{dt} &= \nu k_j v_{*j} + y_j(k_jv_{*j}-b), \quad j=1,\dots,n_2& \\
\frac{dv_{ij}}{dt} &= v_{ij}(r_{ij}-p_ix_i-q_jy_j), \quad i=1,\dots,n_1 \text{ and } j=1,\dots,n_2&
\end{align*}

\citet{Funk2005} has incorporated spatial dynamics into this set of \glspl{ODE} by discretizing the tissue into a 2D plane:

\begin{align*}
\frac{\text{d}T_{i,j}}{\text{d}t} &= \lambda - d T_{i,j} - \beta V_{i,j}T_{i,j} \\
\frac{\text{d}T^*_{i,j}}{\text{d}t} &= \beta V_{i,j} T_{i,j} - \delta T^*_{i,j} \\
\frac{\text{d}V_{i,j}}{\text{d}t} &= N \delta T^*_{i,j} - c V_{i,j} - \frac{m_\text{V}}{8} \sum_{i_0=i-1}^{i+1} \sum_{j_0=j-1}^{j+1} [V_{i,j} - V_{i_0,j_0}]
\end{align*}




\section{Incorporating Mutations}




\section{Treatment}



\section{Parameter Estimation}





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